When I was young, the difference in America's two political parties wasn't so large as it is today. Consequently, to win an election a politician might adjust his views to incorporate more of the "middle views" and fewer of the extreme views of voters.
Today, the difference in our two political parties is so large that to win an election, politicans perceive that they need to win more of the extremist votes, and therefore they move their position further from the center. Republicans become more right-wing and Democrats become more left-wing. This gives us two parties that are farther apart than ever and also gives us the daily bickering that we see between the two major parties in Congress.
That leaves Independent voters in the center of the political spectrum with influence that surpasses what one would expect from their numbers alone. Independent voters have become more important, not because they're going to budge the bulk of either of the two main parties closer to the center, but because throwing their much smaller weight to one candidate can decide the outcome of an election.
Of course, a candidate still must win the votes of the "regular" voters, so he can't change his positions in a major way. But it's to his advantage to gather some Independent voters if possible. The Independent voters today are more important than ever, because Republican voters and Democrat voters are a given—we know how they're going to vote—and so the outcome of an election often depends on getting as many Independent votes as possible.
On the other hand, one could argue that, given that the nation's voting districts have been gerrymandered to hell, it might be in a candidate's best interest to ignore Independent voters and appeal to the extremist voters. Gerrymandering has, in effect, rigged the outcome of our elections to the point where voters can be forgiven for asking if casting a vote has much meaning anymore. Just pick an 'R' or a 'D' and roll the dice.
